• DOI:10.4204/EPTCS.158.7
• Corpus ID: 1296189

Gelfand spectra in Grothendieck toposes using geometric mathematics

@inproceedings{Spitters2013GelfandSI,
  title={Gelfand spectra in Grothendieck toposes using geometric mathematics},
  author={Bas Spitters and Steven J. Vickers and Sander Wolters},
  booktitle={QPL},
  year={2013}
}

• Bas Spitters, S. Vickers, Sander Wolters
• Published in QPL 2 October 2013
• Mathematics

In the (covariant) topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to each unital C*-algebra, A, a topos T(A) of sheaves on a locale and a commutative C*-algebra, a, within that topos. The Gelfand spectrum of a is a locale S in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale S, by noting that the construction of the Gelfand spectrum in a general topos can be described using geometric logic. As a consequence, the spectrum, seen as a bundle, is computed fibrewise.
As a by-product of the geometricity of Gelfand spectra, we find an explicit external description of the spectrum whenever the topos is a functor category. As an intermediate result we show that locally perfect maps compose, so that the externalization of a locally compact locale in a topos of sheaves over a locally compact locale is locally compact, too.